Sure, let's find the square root of 1008 using prime factorization step-by-step:
1. Prime Factorization:
First, we need to break down 1008 into its prime factors.
- Start with the smallest prime number, 2:
[tex]\[
1008 \div 2 = 504
\][/tex]
[tex]\[
504 \div 2 = 252
\][/tex]
[tex]\[
252 \div 2 = 126
\][/tex]
[tex]\[
126 \div 2 = 63
\][/tex]
- Now move to the next smallest prime number, 3:
[tex]\[
63 \div 3 = 21
\][/tex]
[tex]\[
21 \div 3 = 7
\][/tex]
- The remaining number, 7, is already a prime.
We have now:
[tex]\[
1008 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7
\][/tex]
2. Group the Prime Factors:
Next, we group the prime factors in pairs (as we are looking for a square root):
[tex]\[
(2 \times 2) \times (2 \times 2) \times (3 \times 3) \times 7
\][/tex]
This can be rewritten using exponents:
[tex]\[
1008 = 2^4 \times 3^2 \times 7
\][/tex]
3. Halve the Exponent:
To find the square root, we take half of each of the exponents in the factorization (where possible):
[tex]\[
\sqrt{1008} = \sqrt{2^4 \times 3^2 \times 7} = 2^{4/2} \times 3^{2/2} \times \sqrt{7}
\][/tex]
Simplifying the exponents:
[tex]\[
\sqrt{1008} = 2^2 \times 3^1 \times \sqrt{7} = 4 \times 3 \times \sqrt{7}
\][/tex]
Combining these:
[tex]\[
\sqrt{1008} = 12 \times \sqrt{7}
\][/tex]
4. Estimate the Numerical Value:
Finally, we approximate the value of [tex]\(\sqrt{7}\)[/tex], which is roughly 2.64575. So:
[tex]\[
12 \times \sqrt{7} \approx 12 \times 2.64575 = 31.749
\][/tex]
So, the square root of 1008 is approximately [tex]\(31.749\)[/tex]. The detailed steps confirm that the prime factors of 1008 are [tex]\(2, 2, 2, 2, 3, 3, 7\)[/tex]. The half counts product is 12, the remaining part inside the square root is 7, and incorporating this value, we get a final square root of approximately [tex]\(31.749\)[/tex].
